Electron hopping integral renormalization due to anharmonic phonons

Interpretation of some experimental data of temperature-dependent transport properties of a solid material has led to a hypothesis that some renormalization has occurred to the electron hopping integral, making it decrease as temperature increases up to some limit. Such a renormalization may be attributed to the effect of coupling of electrons with anharmonic phonons. Namely, the presence of anharmonic vibration of two adjacent atoms yields an increase of their equilibrium distance as temperature increases, this in turn weakens the electron hopping integral connecting the two atomic sites. Despite that the above qualitative picture is pretty clear, little has been explored on the detail of how such an electron hopping integral evolves as a function of temperature. Motivated by this issue, here, we propose a tight-binding model incorporating the presence of anharmonic phonons and electron-phonon coupling, and solve it using Green function technique. We define the renormalized electron hopping integral from the band width of the interacting system, and perform self-consistent calculations to demonstrate the evolution of the electron hopping integral as a function of equilibrium distance between adjacent atoms and temperature.